The Arbitrage-Free Valuation Framework
READING 26
THE ARBITRAGE-FREE VALUATION FRAMEWORK
EXAM FOCUS
This topic review discusses valuation of fixed-income securities using spot rates and the backward induction methodology in a binomial interest rate tree framework. It explains how embedded options affect the suitability of the binomial model versus the Monte Carlo simulation method and summarises related term-structure models.
Video covering this content is available online.
MODULE 26.1: BINOMIAL TREES, PART 1
LOS 26.a: Explain what is meant by arbitrage-free valuation of a fixed-income instrument.
The arbitrage-free valuation framework applies the law of one price in liquid markets: identical cash flows must have the same price. Valuation methods that are arbitrage-free assign values so that no market participant can earn an arbitrage profit-a future positive, riskless profit achieved with zero initial net investment.
Two common forms of arbitrage opportunity addressed by this framework are:
- Value additivity violation - when the value of a whole security differs from the sum of values of its parts (enabling stripping or reconstitution arbitrage).
- Dominance - when one asset trades at a lower price than another asset with identical cash flows or payoff characteristics.
Example of value additivity arbitrage: a five-year 5% Treasury bond should be worth the same as the portfolio of its coupon and principal strips. If the intact bond trades for less than the strips, one can buy the bond, strip it, and sell the strips for a profit. Conversely, if the strips trade for less than the intact bond, one can buy strips, reconstitute them into the bond, and sell the bond.
EXAMPLE: Arbitrage opportunities
The following information has been collected:
Securities A and B are identical in every respect other than as noted. Similarly, securities C and D are identical in every other respect.
Demonstrate the exploitation of any arbitrage opportunities.
Answer:
1.
Arbitrage due to violation of the value additivity principle:
2.
Arbitrage due to the occurrence of dominance:
Note: The example statement above is retained as given. The precise trading actions and cash flows that produce arbitrage profits depend on the numerical prices and cash-flow schedules of the securities, which are assumed to be provided in the original example data.
LOS 26.b: Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond.
Arbitrage-free valuation of an option-free fixed-rate bond requires discounting each of the bond's future cash flows (each coupon payment and the par value at maturity) using the corresponding spot rate (zero-coupon yield) for the appropriate maturity. The present value is the sum of these discounted cash flows.
Mathematical expression (general form):
Price = Σ_{t=1}^{N} CF_t / (1 + s_t)^t
where CF_t denotes the cash flow at time t and s_t the t-period spot rate (expressed per period consistent with coupon frequency).
EXAMPLE: Arbitrage-free valuation
Sam Givens, a fixed income analyst at GBO Bank, has been asked to value a three-year, 3% annual-pay, €100 par bond with the same liquidity and risk as the benchmark. What is the value of the bond using the spot rates provided in the following?
€ Benchmark Spot Rate Curve:
Answer:
Procedure to compute the value (retain for use when specific spot rates are provided):
- Compute each coupon and the principal cash flow: CF_1 = 3, CF_2 = 3, CF_3 = 103 (for €100 par, 3% coupon).
- Discount each CF_t by the corresponding spot rate: PV_t = CF_t / (1 + s_t)^t.
- Sum PV_1 + PV_2 + PV_3 to obtain the bond price.
While option-free bonds can be valued with a simple spot rate curve, bonds with embedded options require modelling the uncertainty of future rates and resulting cash flows. The binomial interest rate tree framework is one such model; it allows forward rates to vary, which affects the probability of option exercise and the bond's cash-flow pattern.
LOS 26.c: Describe a binomial interest rate tree framework.
The binomial interest rate tree framework assumes that, in each one-period step, the relevant one-period forward rate can move to one of two possible values with equal probability (hence the binomial description). Over multiple periods this produces a lattice of possible interest-rate outcomes, which is called a binomial interest rate tree.
Key features and definitions:
- The tree root, i0, is the current one-period spot rate.
- A node is a point where the short or forward rate for that period is realised; each node has an upper (U) and a lower (L) child node for the next period.
- The rate at node (period 2) reached by moving first down then up equals the rate reached by moving first up then down: i2,LU = i2,UL (this symmetry property makes the tree a lattice).
- Rates at nodes for the same period are related by the assumed volatility; adjacent forward rates are separated by two standard deviations (in continuous terms, a multiplier of e^{2σ} between adjacent nodal rates for that period, depending on model specification).
- The framework is typically specified as a lognormal random walk, which implies: (1) higher volatility when rates are higher and (2) interest rates remain non-negative.
Professor's note: Consistent with the curriculum, the term forward rates is used to denote the expected one-period future spot rates in the tree, recognising they are not identical to market-implied forward rates in all contexts.
Binomial tree diagram
A binomial tree diagram depicts nodes with one-period forward rates at each node. The root node is the current one-period spot rate. Subsequent nodes represent possible one-period forward rates conditional on the sequence of up and down moves to that node.
LOS 26.e: Describe the backward induction valuation methodology and calculate the value of a fixed-income instrument given its cash flow at each node.
VALUING AN OPTION-FREE BOND WITH THE BINOMIAL MODEL
Backward induction is the process of valuing a bond by starting at the terminal period and working backwards period by period to determine the current price. For a bond with N periods, you compute the bond's value at the final-period nodes, then use those values to determine values one period earlier, continuing back to the root.
When probabilities of an up move and a down move are both 50%, the value at a given node is the average of the discounted values of the two possible next-period node values. The discount rate used to discount one period is the one-period forward rate associated with the current node.
General backward induction formula at node t with forward rate f_t (one-period rate applicable at that node):
V_t = (1/2) × [ V_{t+1,U} / (1 + f_t) + V_{t+1,L} / (1 + f_t ) ] + Coupon_t / (1 + f_t)
where V_{t+1,U} and V_{t+1,L} are the bond values at the upper and lower nodes of the next period and Coupon_t is any coupon paid at the current node (depending on coupon convention).
EXAMPLE: Valuing an option-free bond with the binomial model
A 7% annual-coupon bond has two years to maturity. The interest rate tree is shown in the following figure. Fill in the tree and calculate the value of the bond today.
Valuing a 2-Year, 7.0% Coupon, Option-Free Bond
Answer:
Consider the value of the bond at the upper node for Period 1 (V1,U):
Similarly, the value of the bond at the lower node for Period 1 (V1,L) is:
Now calculate V0, the current value of the bond at Node 0.
The completed binomial tree is shown as follows:
Valuing a 2-Year, 7.0% Coupon, Option-Free Bond
Note: The general stepwise procedure above is reproduced in the same order as in the source. Specific numerical rates and the completed numerical tree are included in instructional materials or figures referenced in the original text; when those rates are provided, apply the formulas above to compute the numeric values at each node and at time 0.
MODULE QUIZ 26.1
Use the following information to answer Questions 1 through 5.
Dan Green, CFA, is currently working for FIData, a company specialising in the provision of price data for fixed-income instruments.
Green heads a team that inputs raw data into an instructional area on FIData's website. FIData uses these pages to provide hypothetical bond information along with a description of how to read and interpret the information.
One of FIData's customers has questioned whether some of the data used to demonstrate pricing concepts is correct. The customer's email stated, "The prices of the three bonds used are not consistent with each other and hence may not be accurate. If they were, I would be able to make a significant arbitrage profit (i.e., I could secure the current risk-free rate of return with zero net investment)." Green thinks the customer is misinformed regarding arbitrage gains, but wants to check the data anyway.
The relevant data for three hypothetical risk-free bonds is shown in (<>)Figure 1 given that the benchmark yield curve is flat at 1.50%. (FFPQ is an annual-pay bond.)
Figure 1: Bond Pricing Data
In another area of the company's instructional website, FIData has an explanation of the binomial interest rate tree framework that its analysts use in their valuation process. All of FIData's models populate a binomial interest rate tree assuming interest rates follow a lognormal random walk.
The web page makes two statements regarding the assumptions that underpin the construction and population of such trees:
- Assumption 1: The lognormal model ensures that interest rates are non-negative.
- Assumption 2: The lognormal model ensures a constant volatility of interest rates at all levels of interest rates.
Figure 2: Binomial Interest Rate Tree
FIData's website uses rates in (<>)Figure 2 to value a two-year, 5% annual-pay coupon bond with a par value of $1,000 using the backward induction method.
1.
Green is correct in stating that the customer who sent the email regarding arbitrage gains is misinformed because:
A.
an arbitrage gain always requires a net investment, although this may be small compared to the potential gains.
B.
an arbitrage gain is not constrained by the risk-free rate.
C.
arbitrage gains require a net investment in proportion to returns.
2.
Given the three bonds in (<>)Figure 1, it is possible to make an arbitrage gain by:
A.
selling 1 FFPQ bond and simultaneously purchasing 10 DALO and 10 NKDS bonds.
B.
selling 10 FFPQ bonds and simultaneously purchasing 1 DALO and 1 NKDS bond.
C.
selling 1 DALO bond and 1 NKDS bond and simultaneously purchasing 10 FFPQ bonds.
3.
Which of the following statements regarding the FFPQ bond in (<>)Figure 1 is most likely?
A.
It is priced above its no arbitrage price.
B.
It is priced at its no arbitrage price.
C.
It is priced below its no arbitrage price.
4.
Which of the assumptions regarding the construction of FIData's binomial interest rate trees is most accurate?
A.
Assumption 1 only.
B.
Assumption 2 only.
C.
Neither assumption is correct.
5.
Using the backward induction method, the value of the 5% annual-pay bond using the interest rate tree given in (<>)Figure 2 is closest to:
A.
$900.
B.
$945.
C.
$993.
MODULE 26.2: BINOMIAL TREES, PART 2
LOS 26.d: Describe the process of calibrating a binomial interest rate tree to match a specific term structure.
Constructing a binomial interest rate tree that is useful for valuation requires calibration so the tree reproduces the observed term structure (par curve, spot curve, or zero rates). The calibration process follows these rules:
- Rule 1: The tree must generate arbitrage-free values for benchmark securities - the model must reproduce market prices for the securities used to fit the tree.
- Rule 2: Adjacent forward rates at the same nodal period are separated by two standard deviations (in the model specification); knowing one forward rate in the period allows computation of the rest for that period.
- Rule 3: The middle forward rate (or the midpoint when there is an even number of nodal rates) for a period should be approximately equal to the implied one-period forward rate derived from the benchmark spot rate curve for that period.
Volatility used to generate the tree can be estimated from historical rates or implied from interest rate derivatives.
EXAMPLE: Binomial interest rate tree
Xi Nguyen, CFA, has collected the following information on the par rate curve, spot rates, and forward rates. Nguyen had asked a colleague, Alok Nath, to generate a binomial interest rate tree consistent with this data and an assumed volatility of 20%. Nath completed a partial interest rate tree shown as follows:
Binomial Tree With σ = 20% (One-Year Forward Rates)
Questions:
1. Calculate the forward rate indicated by A.
2. Estimate the forward rate indicated by C.
3. Estimate forward rates B and D.
Answer:
1. Forward rate i1,L is indicated by A and is related to forward rate i1,U given as 5.7883%.
2. Forward rate C is the middle rate for Period 3 and hence the best estimate for that rate is the one-year forward rate in two years f(2,1).
Using the spot rates, we can bootstrap the forward rate:
3. The forward rates B and D are related to C as follows (note that C = i2,LU):
Professor's note: To calculate the value of e^{-0.40} on a TI BA II PLUS calculator, use the following keystrokes: 0.4 [+|-] [2ND] [LN]
LOS 26.f: Compare pricing using the zero-coupon yield curve with pricing using an arbitrage-free binomial lattice.
Valuation using the zero-coupon yield curve (spot curve) is appropriate for option-free bonds: discount each known future cash flow by its corresponding spot rate. For bonds with embedded options, future cash flows depend on whether options (calls or puts) will be exercised, which in turn depends on future interest rates. Therefore, a model that allows forward rates and resulting cash flows to vary-such as a calibrated binomial interest rate tree-is required to value such bonds.
EXAMPLE: Valuation of option-free bond
Samuel Favre is interested in valuing a three-year, 3% annual-pay Treasury bond. He has compiled the following information on Treasury spot rates:
Treasury Spot Rate Curve
Compute the value of the $100 face-value option-free bond.
Answer:
When valuing with spot rates, follow the bond pricing formula presented in LOS 26.b. For bonds with embedded options, because future cash flows depend on interest-rate paths, use a model that allows rates to fluctuate, such as the binomial tree.
EXAMPLE: Valuation of an option-free bond using binomial tree
Samuel Favre is again valuing the same three-year, 3% annual-pay Treasury bond. This time, Favre uses a binomial interest rate tree with specified nodal forward rates (tree rates are given in the instructional figure).
Compute the value of the $100 par option-free bond.
Answer:
Note that the interest rate tree in this example was calibrated to generate arbitrage-free values consistent with the benchmark spot rate curve and therefore produced the same value for the option-free bond as the spot-rate valuation. The calibration ensures the binomial-tree-based valuation equals the spot-curve valuation for option-free bonds used in the calibration.
EXAMPLE: Valuation of option-free bond using pathwise valuation
Samuel Favre wants to value the same three-year, 3% annual-pay Treasury bond. The interest rate tree is the same as before but this time Favre wants to use a pathwise valuation approach.
Compute the value of the $100 par option-free bond.
Answer:
For a three-year bond there are four potential interest rate paths from time 0 through time 3 given a one-period spot known and two future one-period forward rates that can vary: SUU, SUL, SLU, SLL (labels represent the sequence of up and down moves after the initial known spot). Pathwise valuation computes the bond value for each path by discounting each cash flow along the path at the one-period forward rate that applies to that cash flow, and then averages the path values.
Example for one path (notation only, numeric rates depend on the provided tree):
Value for Path 1 = Σ_{t=1}^3 CF_t discounted one period at the path-specific one-period forward rate for each year.
LOS 26.g: Describe pathwise valuation in a binomial interest rate framework and calculate the value of a fixed-income instrument given its cash flows along each path.
Pathwise valuation is mathematically identical to backward induction in a correctly specified binomial tree. For an n-period binomial tree there are 2^{(n-1)} unique paths (excluding the known initial spot). Each path describes the sequence of up and down moves that determine one particular set of forward rates; compute the present value along each path (discounting cash flows at the path-specific one-period rates) and then average across all path values to obtain the current price.
LOS 26.h: Describe a Monte Carlo forward-rate simulation and its application.
Path Dependency
Some securities have path-dependent cash flows-the cash flows at a node depend on the entire rate path that led to that node, not only on the rate level at that node. Mortgage-backed securities (MBS) are a primary example because prepayment behaviour depends on both current and past interest-rate movements.
Example: A mortgage pool formed when rates were 6% that later fell to 4% will have many borrowers refinance at the first 4% dip. If rates later fall again to 4%, the pool will have fewer remaining borrowers able or willing to refinance, so prepayment at the second 4% outcome will be lower. Thus, prepayments are path dependent.
Because the binomial backward-induction method assumes cash flows at a node are independent of the path (i.e., not path dependent), it is inappropriate for MBS valuation. Instead, use a Monte Carlo forward-rate simulation.
Monte Carlo forward-rate simulation:
- Randomly generates a large number of interest-rate paths using a stochastic model with specified volatility and probability distribution.
- Allows modelling of path-dependent cash flows (for example, prepayments that depend on a sequence of rates, not only the current rate).
- Values the security along each simulated path using the path-specific rates and cash flows; the security's value is the average across simulated path values.
- Requires calibration so the simulated paths produce arbitrage-free prices for benchmark securities. Calibration commonly adjusts drift terms (adds/subtracts a constant to all rates) so the expected value from simulations matches market prices - this is called a drift-adjusted model.
- May impose upper/lower bounds or mean-reversion behaviour on simulated rates to reflect empirical tendencies of interest rates.
EXAMPLE: Valuation of option-free bond using Monte Carlo simulation
Samuel Favre is interested in valuing the same three-year, 3% annual-pay Treasury bond. Favre has generated a number of simulated rate paths and will use a drift-adjusted Monte Carlo simulation to value the bond.
Monte Carlo Simulation (Drift-Adjusted)
Answer:
Valuation using the simulated interest-rate paths proceeds by computing the value along each path-discounting cash flows at the one-period rates specified by that path-and averaging these path values. The simulation is calibrated to ensure the average valuation of benchmark securities equals their market prices, yielding an arbitrage-free estimate. Where calibration is exact and the model is correctly specified, the Monte Carlo estimate for option-free bonds will agree with the valuation from spot-rate discounting or a correctly calibrated binomial tree.
MODULE 26.3: INTEREST RATE MODELS
LOS 26.i: Describe term structure models and how they are used.
Term structure models describe the statistical behaviour of interest rates through time and are used to generate possible future term structures for pricing and risk management. Two broad classes are commonly discussed:
- Equilibrium term-structure models - derive the yield curve dynamics from economic fundamentals and often assume a stochastic process for the short rate driven by economic forces.
- Arbitrage-free models - take the observed current term structure as given and construct dynamics that are consistent with no-arbitrage pricing; these models are calibrated to market prices.
Equilibrium Term Structure Models
Equilibrium models seek to explain the term structure based on economic variables. Two single-factor equilibrium models commonly used in curriculum discussions are the Cox-Ingersoll-Ross (CIR) model and the Vasicek model; both use the short-term rate as the single stochastic factor and incorporate mean reversion toward a long-run rate.
The Cox-Ingersoll-Ross (CIR) Model
The CIR model specifies short-rate dynamics with two components: a mean-reversion (drift) term and a stochastic volatility term that depends on the level of the short rate. The functional form (in differential notation) is commonly written as:
dr_t = a(b - r_t) dt + σ √{r_t} dW_t
Interpretation:
- The term a(b - r_t) dt causes the short rate r_t to revert toward long-run mean b at speed a.
- The term σ √{r_t} dW_t is the diffusion (random) component; volatility increases with the level of r_t.
- CIR ensures non-negative interest rates because volatility tends to zero as r_t approaches zero (square-root diffusion).
The Vasicek Model
The Vasicek model is another single-factor mean-reverting model, usually written as:
dr_t = a(b - r_t) dt + σ dW_t
Differences from CIR:
- Volatility is constant (σ) and does not depend on r_t.
- The model permits negative interest rates in some parameter configurations because the diffusion term does not shrink as r_t approaches zero.
Arbitrage-Free Models
Arbitrage-free term-structure models are calibrated to match current market bond prices or the observed yield curve and thus are useful for pricing derivatives and for producing short-rate dynamics consistent with no-arbitrage pricing.
The Ho-Lee Model
The Ho-Lee model is a time-dependent drift model of the short rate derived under no-arbitrage assumptions. In continuous form it can be expressed as:
dr_t = θ_t dt + σ dW_t
Characteristics:
- θ_t is a time-dependent drift function chosen so that the model reproduces the current term structure.
- Assumes constant volatility and produces normally distributed future rates; rates can be negative.
- Calibrated by finding θ_t to match market prices of bonds at each time.
The Kalotay-Williams-Fabozzi (KWF) Model
The KWF model is similar in structure to Ho-Lee but assumes the short rate is lognormally distributed. The continuous form for the logarithm of the short rate is:
d ln(r_t) = θ_t dt + σ dW_t
Characteristics:
- Like Ho-Lee, it is calibrated to market prices via θ_t, but it assumes a lognormal distribution for the short rate (ensuring nonnegative rates).
- Assumes constant volatility applied to the log of the short rate.
Other Models
Multifactor models such as Gauss+ incorporate several rate factors (short-, medium-, and long-term) and may include macroeconomic linkages and mean-reversion in different components. These models can better capture empirical features of the term structure and are used when more precise dynamics are required for pricing or risk management.
MODULE QUIZ 26.2, 26.3
Use the following information to answer Questions 1 through 6.
Farah Dane, CFA, works for Geodesic Investing, a small hedge fund that offers investment services for a handful of clients known personally by the owner, Mike DeGrekker. The fund makes few trades, preferring to wait for what it perceives to be arbitrage opportunities before investing. Last year, the fund managed a return of more than 45%, thanks largely to a single transaction on which the company made a profit of $9.4 million.
The transaction, which DeGrekker described as a "valuation farming exercise" involved simultaneously purchasing a government Treasury and selling the corresponding strips for a higher price than the cost of the Treasury.
Dane is currently using a binomial lattice and a pathwise method to value fixed-income bonds in order to identify potential trading opportunities. She has used the binomial lattice shown in (<>)Figure 1 to value a three-year, annual-pay, 4% coupon risk-free government bond with a par value of $1,000. Her pathwise valuation is also shown.
Figure 1: Binomial Lattice
Dane is not satisfied with this method of valuation and has put together a report for DeGrekker on the use of the Monte Carlo method, which she feels will lead to more accurate valuations. She quotes the following advantages of using Monte Carlo method:
- Advantage 1: The Monte Carlo method will estimate the value of the bond using multiple interest rate paths and hence there is no need to make an assumption about volatility of rates.
- Advantage 2: The method could be applied to get more accurate valuations for securities that are interest rate path dependent.
DeGrekker is resistant to the idea as he is concerned about the amount of computing time the model may require. He accepts, however, that the idea of using many paths is attractive. He concedes that, "increasing the number of paths used in the model increases the statistical accuracy of the estimated value and produces a value closer to the true fundamental value of the security."
1.
Which of the following statements regarding the valuation of an option-free bond using an arbitrage-free binomial lattice is most accurate?
A.
If the binomial lattice is correctly calibrated, it should give the same value for an option-free bond as using the par curve used to calibrate the tree.
B.
The binomial lattice will only produce the same value for an option-free bond as the par curve that was used to calibrate it if the bond is priced at par.
C.
The binomial lattice will only produce the same value for an option-free bond as the par curve that was used to calibrate it if the yield curve is flat.
2.
Which of the following statements most accurately describes the "valuation farming exercise" undertaken by DeGrekker?
A.
DeGrekker used the process of stripping and the law of one price to make an arbitrage gain.
B.
DeGrekker used the process of reconstitution and the principle of no arbitrage to make a risk-free gain.
C.
DeGrekker's profit is not an arbitrage profit as the securities involved are risk free.
3.
Which of the following is most accurate regarding the value Dane would obtain using the backward induction method as opposed to the pathwise valuation method for the bond in (<>)Figure 1?
A.
Both methods would produce the same value.
B.
The pathwise valuation method will give lower values when interest rates are rising because the backward induction method places a higher weighting on earlier cash flows.
C.
The backward induction method will give a different value compared to the pathwise method when the volatility of interest rates is high as the pathwise method uses equal weights.
4.
Dane's pathwise valuation is:
A.
correct.
B.
incorrect, as the correct value is lower than $975.17.
C.
incorrect, as the correct value is higher than $975.17.
5.
Which of the advantages of the Monte Carlo method stated by Dane is most accurate?
A.
Advantage 1 only.
B.
Advantage 2 only.
C.
Neither advantage is correct.
6.
DeGrekker's comment on increasing the number of paths is most likely:
A.
correct.
B.
incorrect in asserting that a larger number of paths will produce an estimate that is statistically more accurate.
C.
incorrect in asserting that a larger number of paths will produce a value closer to the true fundamental value.
7.
When calibrating a binomial interest rate tree to match a specific term structure, which of the following statements is least accurate?
A.
Interest rates in the tree should produce an arbitrage-free valuation for benchmark securities.
B.
Adjacent spot rates at each node of the tree are related by the multiplier e^{2σ}.
C.
The middle forward rate in a period is approximately equal to the implied (from the benchmark spot rate curve) one-period forward rate for that period.
8.
The modern term structure model that is most likely to precisely generate the current term structure is the:
A.
Cox-Ingersoll-Ross model.
B.
Vasicek model.
C.
Ho-Lee model.
KEY CONCEPTS
LOS 26.a
Arbitrage-free valuation leads to a security value such that no market participant can earn an arbitrage profit in a trade involving that security. The valuation is consistent with the principles of value additivity and absence of dominance.
LOS 26.b
Arbitrage-free valuation of fixed-rate, option-free bonds requires discounting each of the bond's future cash flows at the corresponding spot rate (zero-coupon yield) for that cash flow's time to receipt.
LOS 26.c
The binomial interest rate tree framework is a lognormal model with two equally likely outcomes for one-period forward rates at each node. A volatility assumption determines the spread of node rates within each period.
LOS 26.d
A binomial interest rate tree is calibrated such that:
- (1) values of benchmark bonds computed using the tree equal their market prices (no arbitrage),
- (2) adjacent forward rates at any nodal period are two standard deviations apart (model-specified relation), and
- (3) the midpoint forward rate for each period approximates the implied one-period forward rate from the benchmark spot curve.
LOS 26.e
Backward induction is the process of valuing a bond using a binomial interest rate tree by working from terminal nodes back to the present. Each node's value is computed as the discounted average of the next-period node values plus any coupon paid, using the node-specific one-period forward rate.
LOS 26.f
Valuation using the zero-coupon yield curve (spot curve) is suitable for option-free bonds. For bonds with embedded options, where future cash flows depend on future rates, a model that allows forward rates to vary (for example, a calibrated binomial tree) is necessary for arbitrage-free valuation.
LOS 26.g
In pathwise valuation, the value of the bond is the average of the bond values computed along each unique path in the binomial tree. For an n-period binomial tree there are 2^{(n-1)} possible paths (given one known initial spot rate and (n-1) forward-rate decisions).
LOS 26.h
The Monte Carlo simulation method uses pathwise valuation across a large number of randomly generated forward-rate paths. Securities with path-dependent cash flows, such as mortgage-backed securities (due to prepayment options), should be valued using Monte Carlo rather than simple binomial backward induction.
LOS 26.i
Two major classes of term structure models:
- Equilibrium term structure models - model the term structure using economic variables; examples include:
- Cox-Ingersoll-Ross (CIR) model: assumes mean reversion to long-run rate b with speed a; volatility varies with r and the model produces non-negative rates.
- Vasicek model: similar mean reversion but with constant volatility σ; may produce negative rates.
- Arbitrage-free models - take market prices as given and calibrate to them; examples include:
- Ho-Lee: dr_t = θ_t dt + σ dW_t; θ_t is calibrated to match the current term structure; assumes normal distribution of rates and constant volatility.
- Kalotay-Williams-Fabozzi (KWF): d ln(r_t) = θ_t dt + σ dW_t; similar calibration approach but assumes lognormal short-rate dynamics.
ANSWER KEY FOR MODULE QUIZZES
Module Quiz 26.1
1.
B An arbitrage gain is a risk-free profit and hence requires no net investment. The returns therefore are not simply the risk-free rate. As there is no initial investment, the gains cannot be measured as a percentage of initial cost. (LOS 26.a)
2.
B An up-front arbitrage profit of $38.70 can be earned by selling 10 FFPQ bonds short and purchasing 1 DALO and 1 NKDS bond as shown here. (LOS 26.a)
3.
A FFPQ is overpriced. Based on the 1.5% benchmark yield, the other two bonds are correctly priced.
Arbitrage-free price = (PV Year 1 Cash Flow) + (PV Year 2 Cash Flow) (LOS 26.a)
4.
A A lognormal random walk will ensure non-negativity but will lead to higher volatility at higher interest rates. (LOS 26.c)
5.
C The value of the 5%, two-year annual-pay $1,000 par bond is $992.88. (LOS 26.e)
Module Quiz 26.2, 26.3
1.
A A correctly calibrated tree will value the bond at the same price as the par and spot curves used to derive it. (Module 26.2, LOS 26.d)
2.
A DeGrekker purchased the bonds and stripped them into constituent parts before selling them. The strategy involved no initial net investment and yet results in an arbitrage profit. (Module 26.1, LOS 26.a)
3.
A The two methods (backward induction and pathwise valuation) are identical and will always give the same result. (Module 26.2, LOS 26.g)
4.
B This is a tricky question. There are only four possible paths that Dane should have used. The possible paths are UU (Path 1), UD (Path 2), DU (Path 4), and DD (Path 5). Path 3 isn't a valid path. So the value should be the average of the values for Paths 1, 2, 4, and 5. The correct average value is $972.45. (Module 26.2, LOS 26.g)
5.
B Monte Carlo simulation also requires an assumed level of volatility as an input. (Module 26.2, LOS 26.h)
6.
C The larger the number of paths, the more accurate the value in a statistical sense. However, whether the value is closer to the true fundamental value depends on the accuracy of the model inputs. (Module 26.2, LOS 26.h)
7.
B The stated multiplier is correct but it is important to note that the rates given at each node of the tree are forward rates not spot rates. (Module 26.2, LOS 26.d)
8.
C The Ho-Lee model is calibrated by using market prices to find the time-dependent drift term θ_t that generates the current term structure. One drawback of the Vasicek and Cox-Ingersoll-Ross models is that model prices generated by these models generally do not coincide with observed market prices without additional calibration. (Module 26.3, LOS 26.i)
